To expand Wilberd's comment, the ongoing work of R. Bezrukavnikov and I. Mirkovic (following up their joint work with D. Rumynin) takes a more geometric viewpoint. This preprint, now in version 3, addresses the closely related conjectures of Lusztig in 1997-99 on bases in equivariant K-theory: front.math.ucdavis.edu/1001.2562 As in AJS, applications to characteristic $p$ are (so far) dependent on $p$ being "large enough". But the conjectures go beyond restricted Lie algebra representations to those attached to arbitrary nilpotent orbits.
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Jim HumphreysMar 28 '10 at 11:00

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A little more detail about [AJS], which is a 300+ page paper (Asterisque 220, 1994) but has a readable introduction. They work in a graded setting with Lie algebra modules, getting for large enough $p$ and suitably bounded weights a precise comparison with the quantum enveloping algebra of Lusztig at a $p$th root of unity. In the latter case, one combines work of Kashiwara-Tanisaki on the analogue of the Kazhdan-Lusztig Conjecture for affine Lie algebras with work of K-L passing from there to quantum groups. [AJS] relies more on combinatorics than on geometry.
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Jim HumphreysApr 4 '10 at 16:10

Update: This final version of a survey article by Peter Fiebig is an updated version of the last v4 posted on arXiv and is now freely available in PDF format from Bull. London Math. Soc.: blms.oxfordjournals.org/cgi/content/abstract/
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Jim HumphreysAug 30 '10 at 18:57

Apparently, any bound on the largest prime that breaks Lusztig's conjecture must be at least proportional to $n\log n$, and this is probably still far too optimistic; I think it's now believed that you'll need an exponential bound.